The difference is that tax avoidance is
defined as legal and tax evasion
defined as illegal (at least in the US). If it's a legal activity aimed at minimizing taxes, it's avoidance. Likewise, if it's avoidance, it's legal, and if it's not legal, it's not avoidance.
From the IRS manual, section 25.1.1.2.4 .1 & .2 [0] [1]
Avoidance of tax is not a criminal offense. Taxpayers have the right to reduce, avoid, or minimize their taxes by legitimate means. One who avoids tax does not conceal or misrepresent, but shapes and preplans events to reduce or eliminate tax liability within the parameters of the law.
Evasion involves some affirmative act to evade or defeat a tax, or payment of tax. Examples of affirmative acts are deceit, subterfuge, camouflage, concealment, attempts to color or obscure events, or make things seem other than they are.
If I move to Florida or Texas to escape paying state income tax on my earnings, that's avoidance. If I earned the money in Massachusetts and simply decide to not declare and pay tax on it, that's evasion.
What's happening to some of the multi-nationals is that their attempts at structuring their affairs are being judged on the wrong side of the line. While it may not be being called evasion legally in all cases, it's exceeded the bounds of avoidance, resulting in settlements.
[0] - https://www.irs.gov/irm/part25/irm_25-001-001.html#d0e299 (light emphasis mine)
[1] - That section numbering scheme itself gives a clue as to how complex the tax code is.